We investigate the topology of the new Point Source Catalogue
Redshift Survey (PSCz) of {\it IRAS}
galaxies by means of the genus statistic. The survey maps the local
Universe with approximately 15000 galaxies over
84.7 per cent of the sky and provides
an unprecedented number of resolution elements for the topological
analysis. For comparison with the PSCz data we also examine the genus of
large N-body simulations of four variants of the cold dark matter
cosmogony.
The simulations are part of the VIRGO project to simulate
the formation of structure in the Universe.
We assume that the statistical properties of the
galaxy distribution can be identified with those of
the dark matter particles in the simulations.
We extend the standard genus analysis by
examining the influence of sampling noise on the genus curve and
introducing a statistic able to quantify the amount
of phase correlation present in the density field,
the {\it amplitude drop} of the genus compared to
a Gaussian field with identical power spectrum.
The results for PSCz are consistent with the hypothesis of
random phase initial conditions. In particular, no strong
phase correlation is detected on scales
ranging from $10\lu$ to $32\lu$, whereas there is a positive detection
of phase correlation at smaller scales.
Among the simulations,
phase correlations are
detected in all models at small scales, albeit with different strengths.
When scaled to a common normalization,
the amplitude drop primarily depends on the shape of the power
spectrum. We find that
the constant bias standard CDM model
can be ruled out
at high significance
because the shape of its
power spectrum is not consistent with PSCz. The other CDM models with
more large-scale power all fit
the PSCz data almost equally well, with a slight preference for
a high density $\tau$CDM model,
if in addition to the genus the variance of the
density field is considered.